statistical-analysis

> **Source**: [https://github.com/aipoch/medical-research-skills](https://github.com/aipoch/medical-research-skills)

## When to Use

Use this skill when you need to:

1. Choose an appropriate statistical test (e.g., t-test vs. ANOVA vs. non-parametric vs. Bayesian) based on study design and variable types.
2. Validate assumptions before inference (normality, homoscedasticity, linearity, outliers) and decide on remedies when assumptions fail.
3. Run common inferential analyses (hypothesis tests, correlation, regression) and interpret results with effect sizes and uncertainty.
4. Plan studies with a priori power analysis (sample size planning) or run sensitivity analysis after data collection.
5. Write results in APA style with complete reporting elements (test statistic, df, p, effect size, CI, assumption checks).

> For programming model-specific workflows (especially regression variants and custom diagnostics), prefer **statsmodels** directly; this skill focuses on *guided selection, checks, interpretation, and reporting*.

## Key Features

- **Test selection guidance** by research question, design (independent/paired; 2+ groups), outcome type, and distributional properties (see `references/test_selection_guide.md`).
- **Assumption checking and diagnostics** with automated checks and plots (Q–Q, residual plots, boxplots) via `scripts/assumption_checks.py` (see `references/assumptions_and_diagnostics.md`).
- **Frequentist analyses**: t-tests, ANOVA (+ post-hoc), chi-square/Fisher, correlation (Pearson/Spearman), linear/logistic regression with diagnostics.
- **Bayesian alternatives** with posterior summaries and Bayes Factors (see `references/bayesian_statistics.md`).
- **Effect sizes + confidence intervals** for interpretation beyond p-values (see `references/effect_sizes_and_power.md`).
- **APA-style reporting templates** and required reporting elements (see `references/reporting_standards.md`).

## Dependencies

Python (recommended **3.10+**) with:

- `numpy>=1.24`
- `pandas>=2.0`
- `scipy>=1.10`
- `statsmodels>=0.14`
- `pingouin>=0.5`
- `matplotlib>=3.7`
- `pymc>=5.0` (Bayesian workflows)
- `arviz>=0.16` (Bayesian diagnostics/plots)

## Example Usage

The example below is designed to be runnable end-to-end: it generates synthetic data, checks assumptions, runs an independent-samples t-test with effect size, performs power analysis, and prints an APA-style result string.

```python
import numpy as np
import pandas as pd
import pingouin as pg

from statsmodels.stats.power import tt_ind_solve_power

# If your repo provides this module, use it; otherwise comment it out.
from scripts.assumption_checks import comprehensive_assumption_check

# ----------------------------
# 1) Create example dataset
# ----------------------------
rng = np.random.default_rng(7)
n_a, n_b = 50, 52

group_a = rng.normal(loc=75, scale=9, size=n_a)
group_b = rng.normal(loc=69, scale=9, size=n_b)

df = pd.DataFrame({
    "score": np.r_[group_a, group_b],
    "group": ["A"] * n_a + ["B"] * n_b
})

# ----------------------------
# 2) Assumption checks
# ----------------------------
assump = comprehensive_assumption_check(
    data=df,
    value_col="score",
    group_col="group",
    alpha=0.05
)
print("Assumption check summary:")
print(assump["summary"] if "summary" in assump else assump)

# ----------------------------
# 3) Run test + effect size
# ----------------------------
res = pg.ttest(group_a, group_b, correction="auto")  # Welch if needed
t_stat = float(res["T"].iloc[0])
dfree = float(res["dof"].iloc[0])
pval = float(res["p-val"].iloc[0])
d = float(res["cohen-d"].iloc[0])
ci_low, ci_high = res["CI95%"].iloc[0]

# ----------------------------
# 4) Power analysis (planning)
# ----------------------------
n_required = tt_ind_solve_power(
    effect_size=0.5, alpha=0.05, power=0.80, ratio=1.0, alternative="two-sided"
)

# ----------------------------
# 5) APA-style reporting string
# ----------------------------
m_a, sd_a = group_a.mean(), group_a.std(ddof=1)
m_b, sd_b = group_b.mean(), group_b.std(ddof=1)

apa = (
    f"Group A (n = {n_a}, M = {m_a:.2f}, SD = {sd_a:.2f}) and "
    f"Group B (n = {n_b}, M = {m_b:.2f}, SD = {sd_b:.2f}) differed, "
    f"t({dfree:.0f}) = {t_stat:.2f}, p = {pval:.3f}, d = {d:.2f}, "
    f"95% CI [{ci_low:.2f}, {ci_high:.2f}]."
)

print("\nAPA-style result:")
print(apa)

print(f"\nPlanning note: to detect d = 0.50 with 80% power, "
      f"required n per group ≈ {n_required:.0f}.")
```

## Implementation Details

### 1) Test Selection Logic (Conceptual)
Use `references/test_selection_guide.md` as the primary decision aid. The selection is typically driven by:

- **Design**: independent vs. paired/repeated measures; number of groups (2 vs. 3+).
- **Outcome type**: continuous, ordinal, binary/categorical counts.
- **Distribution/assumptions**:
  - approximate normality (overall or within groups),
  - homogeneity of variance (between-group comparisons),
  - linearity and residual behavior (regression),
  - outliers and leverage points.

Common mappings:
- Two independent groups, continuous outcome:
  - normal + equal variances → Student’s t-test
  - normal + unequal variances → Welch’s t-test
  - non-normal/ordinal → Mann–Whitney U
- 3+ independent groups:
  - normal + equal variances → one-way ANOVA
  - unequal variances → Welch/Brown–Forsythe ANOVA
  - non-normal/ordinal → Kruskal–Wallis
- Relationships:
  - continuous–continuous → Pearson (normal) or Spearman (rank/non-normal)
  - continuous outcome + predictors → linear regression
  - binary outcome + predictors → logistic regression

### 2) Assumption Checks and Diagnostics
The automated workflow in `scripts/assumption_checks.py` (referenced in the original documentation) is expected to cover:

- **Outlier detection**: IQR rule and/or z-score thresholds.
- **Normality**: Shapiro–Wilk test plus Q–Q plot.
- **Homogeneity of variance**: Levene’s test plus group boxplots.
- **Linearity (regression)**: residuals vs fitted; optional component-plus-residual checks.

Key parameter:
- `alpha` (default commonly `0.05`): decision threshold for assumption tests.

Recommended remedies (see `references/assumptions_and_diagnostics.md`):
- Normality violations: consider robust/non-parametric tests, transformations, or bootstrap CIs.
- Variance heterogeneity: Welch variants; robust standard errors in regression.
- Linearity violations: transformations, polynomial terms, splines/GAMs.

### 3) Effect Sizes and Uncertainty
Effect sizes should be reported alongside inferential results (see `references/effect_sizes_and_power.md`):

- t-tests: **Cohen’s d** (or Hedges’ g for small samples)
- ANOVA: **partial η²** (or ω² depending on convention)
- correlation: **r** (already an effect size)
- chi-square: **Cramér’s V**
- regression: **R² / adjusted R²**, plus standardized coefficients where appropriate

Always prefer **confidence intervals** (frequentist) or **credible intervals** (Bayesian) to communicate precision.

### 4) Power Analysis
Implemented via `statsmodels.stats.power`:

- **A priori power**: solve for required `n` given target effect size, `alpha`, and desired power.
- **Sensitivity analysis**: solve for detectable effect size given achieved `n`, `alpha`, and desired power.

Avoid “post-hoc power” computed from observed p-values; use sensitivity analysis instead.

### 5) APA-Style Reporting Requirements
Use `references/reporting_standards.md` to ensure inclusion of:

- descriptive statistics (M, SD, n) per group/condition,
- test statistic + df + exact p (or thresholded p where required),
- effect size + CI,
- assumption checks performed and any corrective actions,
- post-hoc procedures and multiple-comparison corrections when applicable.

For Bayesian reporting (see `references/bayesian_statistics.md`), include:
- priors (type and scale),
- posterior summaries and credible intervals,
- Bayes Factor (if used),
- convergence diagnostics (e.g., R̂, ESS) and posterior predictive checks when relevant.